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New trends in computational mechanics: model order
reduction, manifold learning and data-driven
Jose Vicente Aguado, Domenico Borzacchiello, Elena Lopez, Emmanuelle
Abisset-Chavanne, David González, Elías Cueto, Francisco Chinesta
To cite this version:
Jose Vicente Aguado, Domenico Borzacchiello, Elena Lopez, Emmanuelle Abisset-Chavanne, David González, et al.. New trends in computational mechanics: model order reduction, manifold learning and data-driven. 9th Annual US-France symposium of the International Center for Applied Compu-tational Mechanics, Jun 2016, Compiègne, France. �hal-01878190�
New trends in computational mechanics :
mo-del order reduction, manifold learning and
data-driven
Jose Vicente Aguado
1, Domenico Borzacchiello
1, Elena Lopez
1,
Em-manuelle Abisset-Chavanne
1, David Gonzalez
2, Elias Cueto
2,
Fran-cisco Chinesta
11High Performance Computing Institute & ESI GROUP Chair Ecole Centrale de Nantes, 1 rue de la Noe, 44300 Nantes, France @ : Francisco.Chinesta@ec-nantes.fr
2 I3A, Universidad de Zaragoza, Maria de Luna s/n, E-50018 Zaragoza, Spain @ : ecueto@unizar.es
ABSTRACT.Engineering sciences and technology is experiencing the data revolution. In the past
models were more abundant than data, too expensive to be collected and analyzed at that time. However, nowadays, the situation is radically different, data is much more abundant (and ac-curate sometimes) than existing models, and a new paradigm is emerging in engineering sci-ences and technology. This paper retraces some incipient applications based on data within the framework of computational mechanics. Three main topics are addressed in the present work: (i) construction of solution manifolds and its use for interpolating new solutions on the man-ifold; (ii) constructing parametric solutions on the just extracted manman-ifold; and (iii) defining behavior manifolds to perform data-driven simulation while avoiding the use of usual constitu-tive equations.
KEYWORDS: Model Order Reduction, Nonlinear Dimensionality Reduction, Vademecum,
1. Introduction
1.1. The big picture
Engineering sciences and technology, as any other branch of sciences and techno-logy, is experiencing the data revolution. In the past models were more abundant than data, too expensive to be collected and analyzed at that time. However, nowadays, the situation is radically different, data is much more abundant (and accurate sometimes) than existing models, and a new paradigm is emerging in engineering sciences and technology.
Advanced model order reduction techniques allowed moving from data to infor-mation, because in many systems and despite the big amount of data the hidden in-formation was quite reduced, and it was successfully extracted by applying many, nowadays state of the art model reduction techniques (POD, PGD or RB among many other variants).
Prior to solve a given problem the user must introduce the different involved pa-rameters (e.g. material papa-rameters and applied loads) as well as define the domain in which the problem is posed. However the just described procedure has a main han-dicap : it rarely allows proceeding in real-time. In those circumstances the real-time performance required in some applications are compromised. One could think that all these issues could be circumvented with the mere use of more powerful computers. Even if it could be a valuable route, it compromises the accessibility to the appropriate simulation resources of small and medium industries. In order to democratize simula-tion, new solutions are required. A possible alternative consists of calculating offline (using all the needed computational resources and computing time) a parametric solu-tion containing the solusolu-tion of all possible scenario, that is then particularized online using light computational facilities, as deployed devices, tablets or even smartphones, for performing efficient simulation, optimization, inverse analysis, uncertainty propa-gation and simulation-based control, all them under real-time constraints.
Even if someone could think for a while that for constructing the parametric solu-tions just announced it is enough to solve the model at hand for any possible choice of the parameters that it involves, it is clear that such a procedure rapidly fails because it involves a combinatorial explosion (e.g. ten parameters each one taking ten possible values will involve ten to the power of ten possibilities, and 10 parameters taking 10 possible values remains too simplistic in applications of practical interest).
Recently model order reduction opened new possibilities. First, Proper Orthogonal Decompositions – POD – allows extracting the most significant characteristics of the solution, that can be then applied for solving models slightly different to the ones that served to defined the reduced approximation basis, by simply projecting the searched solution onto the extracted reduced approximation basis [CHI16].
Another family of model reduction techniques lies in the used of Reduced Bases constructed by combining a greedy algorithm and "a priori" error indicators. It needs
for some amount off-line work but then the reduced basis can be used on-line for solving different models with a perfect control of the solution accuracy because the availability of error bounds. When the error is inadmissible, the reduced basis can be enriched by invoking again the same greedy algorithm [CHI16].
Finally, Proper Generalized Decomposition methods are based on the use of se-parated representations [LAD85, LAD96, AMM06, AMM10]. Such sese-parated repre-sentations are considered when solving at-hand partial differential equations by em-ploying procedures based on the separation of variables, as described in the next sec-tion.
Advanced clustering techniques not only helps engineers and analysts, they be-come crucial in many areas where models, approximation bases, parameters, ... are adapted depending on the local (in space and time senses) state of the system. They makes possible define hierarchical and goal-oriented modeling.
Machine and manifold learning is also helping for extracting the manifold in which the solutions of complex and coupled engineering problems are living. Thus, uncor-related parameters can be efficiently extracted from the collected data coming from numerical simulations, experiments or even from the data collected from adequate measurement devices. As soon as uncorrelated parameters are identified (constituting the information level), the solution of the problem can be predicted in new points of the parametric space, from adequate interpolations or even more, parametric solu-tions can be obtained within an adequate framework able to circumvent the curse of dimensionality (combinatorial explosion) for any value of the uncorrelated model pa-rameters. Thus, the subtle circle is closed by linking data to information, information to knowledge and finally knowledge to real time decision-making, opening unimagi-nable possibilities within the so-called DDDAS (Dynamic Data Driven Application Systems) that allows even model-free simulations.
All the techniques just referred will be revisited in the present chapter.
1.2. The PGD at glance
Most of the existing model reduction techniques proceed by extracting a suitable reduced basis and then projecting on it the problem solution. Thus, the reduced basis construction precedes its use in the solution procedure, and one must be careful on the suitability of a particular reduced basis when employed for representing the solution of a particular problem. This issue disappears if the approximation basis is constructed at the same time that the problem is solved. Thus, each problem has its associated basis in which its solution is expressed. One could consider few terms in its approximation, leading to a reduced representation, or all the terms needed for approximating the solution up to a certain accuracy level. The Proper Generalized Decomposition (PGD) proceeds in this manner.
When calculating the transient solution of a generic problem u(x, t) we usually consider a given basis of space functions Ni(x), i = 1, · · · , N , the so-called shape
functions within the finite element framework, and approximate the problem solution as u(x, t) ≈ N i=1 ai(t)Ni(x), (1)
that implies a space-time separated representation where the time-dependent coeffi-cientsai(t) are unknown at each time (when proceeding incrementally) and the space
functions Ni(x) are given "a priori", e.g. polynomial basis. POD and Reduced Bases
methodologies consider a reduced basis φi(x) for approximating the solution instead
of using the generic functionsNi(x). The former are expected to be more suitable for
approximating the problem at-hand. Thus, it results
u(x, t) ≈
R
i=1
bi(t)φi(x), (2)
where in general R N . Again (2) represents a space-time separated representa-tion where the time-dependent coefficient must be calculated at each time during the incremental solution procedure.
Inspired from these results one could consider the general space-time separated representation u(x, t) ≈ N i=1 Xi(x)Ti(t), (3)
where now neither the time-dependent functions Ti(t) nor the space functions Xi(x)
are "a priori" known. Both will be computed on-the-flight when solving the problem. As soon as one postulate that the solution of a transient problem can be expressed in the separated form (3) whose approximation functions Xi(x) and Ti(t) will be
determined during the problem solution, one could make a step forward and assume that the solution of a multidimensional problem u(x1, · · · , xd) could be found in the
separated form u(x1, x2, · · · , xd) ≈ N i=1 Xi1(x1)Xi2(x1) · · · X d i(xd), (4)
and even more, expressing the 3D solution u(x, y, z) as a finite sum decomposition involving lower dimensional functions
u(x, y, z) ≈ N i=1 Xi(x)Yi(y)Zi(z), (5) or u(x, y, z) ≈ N i=1 Xi(x, y)Zi(z), (6)
and the solution of a parametric problemu(x, t, p1, · · · , p℘) as u(x, t, p1, · · · , p℘) ≈ N i=1 Xi(x)Ti(t) ℘ k=1 Pik(pk). (7)
The performances of all these separated representations are quite impressive in many cases.
For a review on such techniques and their applications in engineering sciences the interested reader can refer to [CHI10, CHI11, CHI13a, CHI13b] and the nu-merous references therein concerning the space-time / space-frequency decomposi-tion [BOU97, LAD99, AMM07, AMM11, BOU13, RIO13, BAR14], space separa-tion [BOG12, VID12, VID13, BOG14, BOR15] and parametric solusepara-tions [CHI13b, HEY13, AMM14, NER15], allowing real-time simulations [CHI13b, GON14, GON15], optimization [GHN11, CHI13b, CHI14, BOR16, AGU17], simulation-based control [GHN12, GON12, CHI13b, AGU15], uncertainty propagation [NOU08, NOU09a, NOU09b, NOU10] or multi-scale upscaling [LAM10, NER10, CRE13].
In all the applications referenced above the choice of parameters was a simple matter : material or process parameters, boundary conditions, etc ... However, in other applications the extraction of uncorrelated model parameters is not an easy task, as it is for example the case when addressing shape parametrization or the description of microstructures. These issues will be addressed in what follows.
2. Constructing slow manifolds
It is well known that microstructures or shapes do not allow simple reduced des-criptions. The main question is not if microstructures or shapes define or not slow manifolds, the question is if they can or not be parametrized, i.e. represented, from slow manifolds. The same issue applies in visualization of high-dimensional data.
2.1. From Principal Component Analysis – PCA – to Kernel Principal Component Analysis – kPCA
Let us considerD observed variables defining the vector X ∈ RD. These are com-monly referred to in the MOR literature as snapshots. We assume that these variables are therefore not uncorrelated and, notably, that there exists a linear transformationW defining the vector Y ∈ Rd, where d < D represents the unknown so-called latent variables, according to
X = WY. (8)
The transformation W, D × d, is assumed to verify the orthogonality condition
WTW = I
d, whereIdrepresents thed × d-identity matrix (WWT is not necessarily
We assume the existence ofM different snapshots X1, . . . , XM, that can be stored
in the columns of the D × M matrix X. The associated d × M reduced matrix Y contains the associated vectorsYi,i = 1, . . . , M .
PCA is able to calculate both d —the necessary number of members in the basis of the reduced-order subspace— and the transformation matrixW. PCA proceeds by guaranteeing maximal preserved variance and decorrelation in the latent variable set. From a statistical point of view, therefore, it can be assumed that the latent variables are uncorrelated (no linear dependencies among them) or mutually orthogonal, thus constituting a basis. In practice, this means that the covariance matrix ofY, defined as
Cyy = E{YYT}, (9)
is diagonal. Thus, we consider
Cxx = E{XXT} = E{WYYTWT} = WE{YYT}WT = WCyyWT, (10)
that by pre-multiplying and post-multiplying byWT and W respectively, and taking into account thatWTW = I, leads to
Cyy = WTCxxW. (11)
The covariance matrixCxx can then be factorized by applying the singular value
decomposition,
Cxx = VΛVT, (12)
withV containing the orthonormal eigenvectors and Λ the diagonal matrix containing the eigenvalues (non-negative real numbers), assumed in descending order.
Substituting the factorized expression of the covariance matrix (12) into Eq. (11) it results
Cyy = WTVΛVTW. (13)
This equality holds only when the d columns of W are taken collinear with d columns ofV.
From a geometrical point of view, the columns ofV indicate the directions in RD that span the subspace of the latent variables. We illustrate this interpretation in Fig. 1 where at left we can appreciate points that apparently belongs to R2, however, it is easy to see that all this point belongs to a slow one-dimensional manifold. PCA find an alternative coordinate system given byV (axes in red) in which all these points are described from a single coordinate.
Nonlinear methods are often more powerful than linear ones, because the connec-tion between the latent variables and the observed ones may be much richer than a simple matrix multiplication. This situation is sketched in Fig. 2 where it can be noti-ced that no-rotation allows to extract the one-dimensional slow manifold. Thus, PCA indicates that the different points belongs to a two-dimensional space, with the risk of
Figure 1. Geometrical interpretation of PCA
Figure 2. PCA limits in presence of strongly-nonlinear manifolds
concluding that the closest point (using the 2D euclidean distance) to the red point is in fact one that is very far from it when using the more appropriate geodesic distance on the one-dimensional slow manifold. Thus, the extraction of the slow manifold is compulsory and PCA is unable to accomplish the job.
These limitations justify the use of nonlinear dimensionality reduction techniques, as the local-PCA (lPCA), the kernel-PCA (kPCA) or the Locally Linear Embedding (LLE).
Local PCA applies standard PCA locally, that is, at each data-point and its closest neighbors. It is sketched in Fig. 3. The main issue related to its practical implementa-tion is the alignment of the local bases unfolding the slow manifold, as discussed in many papers, e.g. [ZHA04].
Figure 3. Scketch of local-PCA
2.2. Kernel Principal Component Analysis (kPCA)
PCA works with the sample covariance matrix, XXT. On the contrary, kPCA works with the matrix of pairwise scalar products that defines the Gram matrix S = XTX as it is also the case of Multidimensional Scaling (MDS) [LEE07]. In its
clas-sical version, MDS preserves pairwise scalar products instead of pairwise distances (both are closely related). It proceeds from
S = XTX = YTWTWY = YTY,
(14)
whose eigenvalue decomposition results
S = UΛUT = UΛ1/2 Λ1/2UT= Λ1/2UTT Λ1/2UT
, (15) from which it results
Y = Id×MΛ1/2UT. (16)
The idea behind kernel-PCA methods is simple : data not linearly separable in D dimensions, could be linearly separated if previously projected to a space in Q > D dimensions. Thus, surprisingly, kPCA begins by projecting the data to an even higher dimensional space. One the biggest advantages of this technique is that there is no need to explicitly determine the analytical expression of the mapping.
The symmetric matrix related to the mapped snapshotsZTZ has to be decomposed in eigenvalues and eigenvectors, after performing to the vectors involved in Z the double centering [LEE07].
Now, the eigenvalue-eigenvector decomposition can be performed according to
ZTZ = UΛUT
, (17)
from which it results
Y = Id×MΛ1/2UT. (18)
It is worth noting that the previous procedure only needs scalar products in the intermediate spaceRQ. The Mercer’s theorem allows computing such scalar products in the original spaceRD, by using the so-called kernel-trick. There are many possible choices [LEE07] and then the model calibration becomes its principal advantage and at the same time its main drawback.
2.3. Locally Linear Embedding – LLE
First we assume the existence of M multi-dimensional data Xm,m = 1, . . . , M ,
defined in a space of dimensionD, i.e. Xm ∈ RD. LLE proceeds as follows [ROW00] :
– Each point Xm, m = 1, . . . , M is linearly reconstructed from its K-nearest
neighbors. In principle K should be greater that the expected dimension d of the un-derlying embedded slow manifold and the points should be close enough to ensure the validity of the linear approximation. In general, a large-enough number of neighbors K and a large-enough sampling M ensures a satisfactory reconstruction. For each point Xmwe can write the locally linear data reconstruction as :
Xm =
i∈Sm
WmiXi, (19)
where Wmi are the unknown weights and Sm the set of the K-nearest neighbors of
Xm. As the same weights appears in different locally linear reconstructions, the best
compromise is searched by looking for the weights, all them grouped in vector W, that minimize the functional
F(W) = M m=1 Xm− M i=1 WmiXi 2 (20)
where here Wmi is zero if Xi does not belong to the set of K-nearest neighbors of
Xm. The minimization of F(W) allows to determine all the weights involved in all
the locally linear data reconstruction.
– We suppose now that each linear patch aroundXm,∀m, is mapped into a lower
dimensional embedding space of dimensiond, d D. Because of the linear mapping of each patch, weights remain unchanged. The problem becomes the determination of the coordinates of each point Xm when it is mapped into the low dimensional space,
For this purpose a new functional G is introduced, that depends on the searched coordinates Y1, . . . , YM : G(Y1, . . . , YM) = M m=1 Ym− M i=1 WmiYi 2 , (21)
where now the weights are known and the reduced coordinates Ym are unknown.
The minimization of functional G results in a M × M eigenvalue problem whose d-bottom non-zero lowest eigenvalues define the set of orthogonal coordinates in which the manifold is mapped.
The use of LLE exhibits some weakness, the first related to the use of euclidian dis-tances, even if other choices could be considered. The second is related to a covariance normalization considered when solving the second eigenproblem above. Alternatives, like the t-SNE [MAA08] allows circumventing the last difficulty.
2.4. Discussion
The main advantage of local-PCA is that it allows extracting the real local reduced dimensionality and the fact of having a real geometrical transformation allowing not only extracting the embedded manifold but also to map points outside the slow mani-fold. Moreover, PCA-based transformations preserve distances, where other nonlinear dimensionality reduction strategies fail to accomplish it.
3. Manifold-learning-based computational mechanics
Imagine different physical systems characterized by vectors Xm ∈ RD, whose
associated solutions (of the physical problem at hand) are dented byTm. By applying
the LLE (among other possible choices) the slow manifold is extracted. Now, when considering a new physical system, characterized byX, by applying the LLE it results its image Y on the manifold, with the weights associated to its reconstruction, i.e.
Y =
i∈S(X)
WiYi, (22)
whereS(X) represents the set of the K-nearest neighbors of X. Now, a prediction of the problem solutionT writes
T =
i∈S(X)
WiTi. (23)
This strategy was successfully considered in [LOP16] for addressing models invol-ving parametrized microstructures and shapes. However, there is a strong assumption
in the rationale just described. The neighbors and their associated weights in the para-metric space are consider to interpolate the solution.
A more accurate approach consists of calculating the parametric solution within for example the PGD framework,
T (x, t, y1, · · · , yd) ≈ N i=1 Xi(x)Ti(t)Yi1(y1) · · · Y d i(yd), (24)
where here x denotes the space coordinates involved in usual models and their as-sociated partial differential equations, t the time involved in transient models and yj
are the latent variables grouped in vector Y (defining the slow manifold). This pro-cedure was successfully applied in [GON16] for addressing the same problems that were addressed in [LOP16].
4. Data-Driven simulations
We consider mechanical tests conducted on a perfectly elastic material, in a spe-cimen exhibiting uniform stresses and strains. For a while, we do not consider issues related to data generation and collection, they will addressed later. More complex be-haviors were addressed in [IBA16]. Thus, forM randomly applied external loads, we assume ourselves able to collect M couples (σm, m), m = 1, . . . , M. Each
stress-strain couple could be represented as a single pointPm in a phase space of dimension
D = 12 (the six distinct components of the stress and strain tensors, respectively). In the sequel Voigt notion will be considered, i.e. stress and strain tensors will be repre-sented as vectors and consequently the fourth-order elastic tensor reduces to a 6 × 6 square matrix.
Each vector Pm thus defines a point in a space of dimension D and, therefore,
the whole set of samples represents a set of M points in RD. We conjecture that all these points belong to (or can be embedded into) a certain low-dimensional manifold embedded into the high-dimensional space RD allowing for a nonlinear dimensiona-lity reduction as discussed in [IBA16]. In what follows we proceeds without such a dimensionality reduction and consider the simplest strategy proposed and discussed in [IBA16]. We consider locally linear approximations, that allow writing
Pm = M
i=1
WmiPi, (25)
with Wmi = 0 if i /∈ Sm (set containing the K-nearest neighbors of Pm). By
mini-mizing the functional
H(C) =
i∈Sm
(σi− Ci)2. (26)
4.1. Data-based weak form
From the just identified locally linear behaviorC(P) one could apply the simplest linearization technique operating on the standard weak form
Ω
∗(x) : σ(x) dx =
ΓN
u∗(x) · t(x) dx, (27)
where at each point, from the stress-strain couple at position x, P(x), the locally linear behaviorC(P(x)) can be obtained (in practice at the Gauss points used for the integration of the weak form) that allows us to write, using Voigt notation
Ω
∗(x) · (C(x)(x)) dx =
ΓN
u∗(x) · t(x) dx. (28)
This allows, in turn, to compute the displacement field and from it, to update the strain and stress fields, and compute again the locally linear behavior. The process continues until convergence. The discretization related to other two alternative des-criptions was deeply considered in [IBA16].
4.2. Constructing the constitutive manifold
In the sequel we consider perfectly elastic isotropic behavior, i.e. inelastic (irre-versible) deformations are neglected, and also assume small displacements and defor-mations. We consider the mechanical specimen occupying the domain Ω ∈ R3, of boundary Γ ≡ ∂Ω with prescribed displacements on ΓD,u(x ∈ ΓD) = ug, without
loss of generality assumed vanishing, i.e.ug = 0, and prescribed tractions in the
com-plementary boundary ΓN (Γ = ΓD ∪ ΓN), σ · n|x∈ΓN = t. Boundary ΓN is at its
turn decomposed in the free-traction region ΓfN where t = 0 and the remaining part Γt
N where non null external tractions,t = 0, apply.
We also ignore mass and inertia contributions to the mechanical state and assume that in absence of external tractions, i.e. whent = 0, the mechanical part remains free of strains and stresses, i.e. = 0 and σ = 0 respectively.
4.2.1. Linear elastic behavior
The external tractiont is applied, from which the equilibrium reads
Ω
∗(x) · σ(x) dx =
ΓN
u∗(x) · t dx. (29)
Taking into account the strain- and stress-free reference configuration previously discussed, the problem can be expressed in the incremental form
ΩΔ
∗(x) · Δσ(x) dx =
ΓN
or introducing the linear behavior ΩΔ ∗(x) · (CΔ(x)) dx = ΓN Δu∗(x) · t dx, (31)
where the tangent matrix C is unknown, but because the linear elastic behavior re-mains constant everywhere in the domain. Using a parametrization of symmetric6×6 matrices (the more general one making use of canonical matrices with a single nonzero entry, taking a unit value) we can write
C =
K
i=1
αiKi, (32)
with coefficientsαiunknown. When using canonical matrices K = 21, and any
sym-metric matrix can be written from a linear combination of those21 canonical matrices by considering adequate coefficientsαi,i = 1, · · · , 21.
By introducing this tangent matrix representation into the equilibrium weak form it results ΩΔ ∗(x) · K i=1 αiMi Δ(x) dx = ΓN Δu∗(x) · t dx, (33)
whose discrete form reads
ΔU∗· K i=1 αiKi ΔU = ΔU∗ · T, (34)
withKithe rigidity matrices corresponding to the canonical behaviors.
We assume that local displacement and their associated strains are accessible on a certain region of the domain whose associated degrees of freedom are indicated with the superscript •O. Thus, making use of a partition of the displacements ΔUO and ΔUH referring to the observable and hidden displacements respectively, the previous
discrete nonlinear system can be solved to compute the unknown displacementsΔUH and material coefficientsαi.
When considering a linear behavior the resulting displacements, strains and stresses can be easily derived fromU ≡ ΔU = (ΔUH, ΔUO)T by considering
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ u(x) = N i=1 UiNi(x) = ∇Su σ = C , (35)
whereN is the number of nodes considered for approximating the displacement field
u(x) and Ni(x) the associated shape functions. ∇S• denotes the symmetric
com-ponent of the gradient operator andC results from αi
C =
K
i=1
αiKi. (36)
4.2.2. Nonlinear elastic behavior
In the nonlinear case, when performing homogeneous tests, it suffices applying the external load incrementally and at each step, being the behavior the same everywhere in the tested coupon, the behavior will be identified incrementally. However, such a procedure do not allow exploring the whole strain-stress space, there are not testing facilities able to prescribe any complex multi-axial strain. In our recent works we pro-pose an alternative data-driven inverse procedure that more than using homogeneous tests, exploits complex tests to cover as much as possible the behavioral manifold. This approach combines an incremental loading and a clustering to identify the candi-date points where the behavior should be upcandi-dated from the inverse procedure at each loading step.
5. Data-driven upscaling of viscous flows in porous media
Isothermal flows of complex fluids in complex microstructures can be simulated by solving the momentum and mass balance equations and a suitable rheological consti-tutive model. For inertialess incompressible flows, these balance equations read,
∇ · σ = 0, (37)
and
∇ · v = 0, (38)
respectively. Here,σ is the Cauchy stress tensor and v the velocity field, both defined at time t at each point within the fluid domain Ωf. When considering porous media,
the domain Ω is assumed fully saturated, with the fluid phase occupying the region Ωf whereas the remaining partΩs = Ω − Ωf is occupied by a solid phase assumed at
rest.
An appropriate constitutive equation must be postulated to describe the fluid’s rheology. There are many possible choices, the most usual ones being related to New-tonian and the generalized-NewNew-tonian summarized below.
For a Newtonian fluid, the constitutive equation reads
σ = −pI + τ = −pI + 2ηD, (39)
where p is the pressure field that can be interpreted as the Lagrange multiplier asso-ciated with the incompressibility constraint,I is the identity tensor, τ the extra-stress
tensor,η the constant fluid viscosity and D the rate of strain tensor, i.e. the symmetric part of the velocity gradient, 2D = ∇v + (∇v)T.
For a generalized-Newtonian fluid, the constitutive equation (39) remains formally unchanged but now the viscosity η depends on the effective strain rate ˙γ usually ex-pressed from the second invariant of the rate of strain tensor, i.e. ˙γ = √2D : D. The simplest of such models is the power-law (shear-thinning) viscosity given by
η = κ ˙γn−1, (40)
whereκ and n are the consistency and power-law index, respectively. The value n = 1 corresponds to a Newtonian fluid.
5.1. Upscaling Newtonian and generalized-Newtonian fluids flowing in porous media
The upscaling procedure was deeply addressed in our former works [LOP15, LOP16b, AMM16] to handle Newtonian, generalized-Newtonian fluid and quasi-Newtonian fluids.
For all these fluids, the flow model is solved in the representative volume ω(X) located at potionX ∈ Ω, where two phases coexist, i.e. the fluid phase occupying the domain ωf(X) and the solid phase, assumed rigid and at rest, occupying the region
ωs(X), with ωf(X) ∪ ωs(X) = ω(X) and ωf(X) ∩ ωs(X) = ∅. The flow model
consists of the mass and momentum balances complemented by the constitutive equa-tion discussed in the previous secequa-tion,
⎧ ⎨ ⎩ ∇ · σ = 0 ∇ · v = 0 σ = −pI + 2ηD = −pI + τ , (41)
with the viscosity η constant in the case of Newtonian fluids or depending on the effective strain rate in the case of generalized-Newtonian fluids. The flow model above is complemented with the boundary conditionv(x ∈ ∂ω(X)) = V, where V comes from the macroscopic flow problem.
The solution of the flow problem (41) allows calculating the velocity fieldv(x ∈ ωf(X)), from it the strain rate D(x ∈ ωf(X)), the local viscosity η(x ∈ ωf(X))
and finally the extra-stress tensor τ (x ∈ ωf(X)), fields that allow calculating the
dissipated power in the RVE,DP(V; X), associated with the prescribed macroscopic velocityV on its boundary ∂ω
DP(V; X) =
ωf(X)
σ(x) : D(x)dx, (42)
with the specific microscopic dissipation DPm obtained by dividing DP given by (42) by the RVE volume|ω(X)|.
Obviously, being the model purely viscous it only involves dissipated power, and consequently the effective macroscopic model should account for that dissipated po-wer. When considering the Darcy’s model, the specific macroscopic dissipated power DPM
reads
DPM(∇P, V) = ∇P · V,
(43) where by equating the micro and macro dissipations, it results
DPm = ∇P |
X· V(X), (44)
or by assuming an effective permeability Kef f(X)
∇P |X = K−1ef f(X)V(X), (45)
from which it finally results
DPm(V; X) = K−1
ef f(X) : (V(X) ⊗ V(X)). (46)
The previous expression constitutes a constructive definition of the effective per-meability. For calculating the last it suffices taking the second derivatives ofDPm(V) related to the microstructure existing at locationX,
K−1
ef f(X) =
d2DPm(V; X)
dV2 . (47)
In the case of a Newtonian fluid, the velocity, strain-rate and stress fields scale linearly with the prescribed velocity on the RVE boundary and consequently the dissi-pated power scales with the square of the velocity, leading to a constant permeability, as deeply discussed in [LOP15].
6. Conclusions
We are not at the beginning of the end, but at the end of the beginning ! Data is expected enriching modeling approaches and even replacing too poor models in order to improve predictions accuracy. The big amount of collected data, including synthetic data generated from simulations, should perform bringing data, information, knowledge and decision making, operating under real-time constraints.
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